Limiting set of second order spectra

Abstract

Abstract. Let M be a self-adjoint operator acting on a Hilbert space H. A complex number z is in the second order spectrum of M relative to a finitedimensional subspace L ⊂ Dom M 2 iff the truncation to L of (M − z) 2 is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on M, the uniform limit of these sets, as L increases towards H, contain the isolated eigenvalues of M of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

On approximation of the eigenvalues of perturbed periodic Schrödinger operators

“…The core idea of the quadratic projection method lies in the fact that Spec(A) can be well estimated if one knows the points of Spec(P L ) which are "close" to the real line, see Corollary 2.5 and Theorem 2.7 below. In [Sh], [LeSh] and [Bo1], Spec(P L ) is called the second order spectrum of A relative to L. This set was first studied in connection with the spectrum of A in [Da], where the name originated.…”

On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators

“…The literature is vast on spectral approximation and we can only refer to a subset here. For selected papers and books we consider to be important and related to the topic of this paper we refer to [9,11,10,26,32,15,14,41,19,18,28,42,47] for a functional analysis exposition and [50,48,25,39] for a more applied mathematical treatment.…”

Quadratic projection methods for approximating the spectrum of self-adjoint operators

“…In our final example we consider a rank one perturbation of a multiplication operator, examples of this type have been considered previously; see Boulton (2006), Boulton (2007), Boulton & Strauss (2007), Davies & Plum (2004), Levitin & Shargorodsky (2004). EXAMPLE 3.5 On the space L 2 [−π, π] with orthonormal basis ψ n (x) = (2π) −1/2 exp(inx), we consider the operator A defined as follows…”

“…Thus without additional information about the structure of Spec(A) we are unable to rely on the Galerkin method. A quadratic version of (1.1) has recently been studied and has the advantage of pollution-free approximation; see Boulton (2006), Boulton (2007), Boulton & Strauss (2007), Boulton & Levitin (2007), Davies (1998), Levitin & Shargorodsky (2004), Shargorodsky (2000).…”

“…If A is unbounded see Boulton (2006) and Boulton & Strauss (2007) for sufficient conditions on sequences (L k ) k∈N to ensure (1.10). It follows from (1.9) that…”